Definition:Injection/Graphical Depiction

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Diagrammatic Presentation of Injection on Finite Set

The following diagram depicts an injection $f$ from $S$ into $T$:

$f: S \to T$

where $S$ and $T$ are the finite sets:

\(\ds S\) \(=\) \(\ds \set {a, b, c}\)
\(\ds T\) \(=\) \(\ds \set {p, q, r, s}\)

and $f$ is defined as:

$f = \set {\tuple {a, p}, \tuple {b, s}, \tuple {c, r} }$


Thus the images of each of the elements of $S$ under $f$ are:

\(\ds \map f a\) \(=\) \(\ds p\)
\(\ds \map f b\) \(=\) \(\ds s\)
\(\ds \map f c\) \(=\) \(\ds r\)
Injection.png
$S$ is the domain of $f$.
$T$ is the codomain of $f$.
$\set {p, r, s}$ is the image of $f$.


The preimages of each of the elements of $T$ under $f$ are:

\(\ds \map {f^{-1} } p\) \(=\) \(\ds \set a\)
\(\ds \map {f^{-1} } q\) \(=\) \(\ds \O\)
\(\ds \map {f^{-1} } r\) \(=\) \(\ds \set c\)
\(\ds \map {f^{-1} } s\) \(=\) \(\ds \set b\)


Note that $f$ is injective but not surjective:

$\map {f^{-1} } x$ is a singleton for all $x \in \Img f$

but:

$\map {f^{-1} } q = \O$


Sources